Finite Group Databasebeta

← #25939

Subgroups — #25939

Order 256.

641 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 641 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#489C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic
4#2592C211yesmaximal, characteristic, complemented
5#2024C212yesmaximal, complemented
6#2023C212yesmaximal, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal
10#1877C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C23×Q8—21no—
13C22×SD16C2211yescharacteristic
14C22×D8C2211yescharacteristic
15D4×Q8—24no—
16D4×Q8—24no—
17C2×C4⋊Q8C2212yes—
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8—22no—
21C2×C22⋊Q8—21no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2212yes—
24C2×C4×D4C2211yescharacteristic
25#489—24no—
26#463—24no—
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#397—21no—
35#393—21no—
36#391C2212yes—
37#390C2212yes—
38#388—22no—
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—28no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—48no—
49C22×Q8D411yescharacteristic
50C22×Q8C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C22×D4C2311yescharacteristic
53C23×C4—22no—
54C23×C4—21no—
55C2×SD16—22no—
56C2×SD16—24no—
57C2×D8—22no—
58C2×D8—24no—
59C22×C8C2311yescharacteristic
60C22×C8C2311yescharacteristic
61C22×C8C2312yes—
62C4⋊Q8—24no—
63C4⋊Q8—24no—
64C4⋊Q8—22no—
65C4⋊Q8D414yes—
66#124—22no—
67#124—22no—
68C22⋊Q8—28no—
69C22⋊Q8—216no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×D4—24no—
73C4×D4—24no—
74C4×D4—22no—
75C4×D4D414yes—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4D412yes—
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2312yes—
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C42—21no—
90C2×C42C2312yes—
91C2×C42C2311yescharacteristic
92C4⋊C8—22no—
93#103—22no—
94#103—24no—
95#102—24no—
96#102—22no—
97#102—24no—
98#97—22no—
99#95—21no—
100#95—22no—
101#95—22no—
102C24—21no—
103C24—21no—
104C2×Q8—28no—
105C2×Q8—22no—
106C2×Q8—24no—
107C2×Q8—24no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—44no—
113C2×Q8—44no—
114C2×Q8—432no—
115C2×Q8—44no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4C2×D412yes—
123C2×D4#4114yes—
124C22×C4—28no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D412yes—
137C22×C4C2411yescharacteristic
138SD16—44no—
139D8—44no—
140C2×C8—22no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C4⋊C4—28no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4C2×D414yes—
156C4⋊C4C2×D412yes—
157C22⋊C4—28no—
158C22⋊C4—24no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C22⋊C4—44no—
162C22⋊C4—82no—
163C42—22no—
164C42—22no—
165C42—81no—
166C42C2×D412yes—
167C42C2×D412yes—
168C23—22no—
169C23—24no—
170C23—42no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—42no—
175C23—42no—
176C23—41no—
177C23—41no—
178C23C22×D411yescharacteristic
179Q8—28no—
180Q8—48no—
181Q8—416no—
182Q8—41no—
183Q8—41no—
184Q8—42no—
185D4—22no—
186D4—24no—
187D4—41no—
188D4—41no—
189D4—42no—
190D4#12414yes—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—48no—
202C2×C4—416no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4#12012yes—
212C2×C4#12012yes—
213C2×C4AGL(1,ℤ/8)14yes—
214C2×C4C22×D411yescharacteristic
215C2×C4C22×D411yescharacteristic
216C2×C4C2×#4112yes—
217C2×C42−1+412yes—
218C8—41no—
219C8—41no—
220C8—42no—
221C22—24no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—42no—
226C22—82no—
227C22—82no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22C2×#12011yescharacteristic
232C22#54812yes—
233C22C2×AGL(1,ℤ/8)12yes—
234C22#57711yescharacteristic
235C22#57811yescharacteristic
236C4—22no—
237C4—24no—
238C4—22no—
239C4—44no—
240C4—42no—
241C4—42no—
242C4—82no—
243C4—81no—
244C4C2×#12411yescharacteristic
245C4#53611yescharacteristic
246C4#54812yes—
247C2—44no—
248C2—161no—
249C2#243611yescharacteristic
250C2#300311yescharacteristic
251C2#301311yescharacteristic
252C2#336712yes—
253C2#337312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×D4×Q8C2semidirect
2. C2×#489C2semidirect
4. #2592C2semidirect
5. #2024C2semidirect
6. #2023C2semidirect
7. #1987C2semidirect
8. #1985C2semidirect
10. #1877C2semidirect
11. #1762C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×D4×Q82
C2×#4892
C2×#4632
#25922
#20242
#20232
#19872
#19852
#18792
#18772
#17622